Relation between AM GM and HM

AM, GM and HM

Relation between AM, GM and HM can be derived with the basic knowledge of progressions or Mathematical sequences. An array or collection of objects in a specified pattern in Mathematics is called a Mathematical Sequence. A sequence is also referred to as a progression. The three most popular types of sequences are Arithmetic sequence, geometric sequence and harmonic sequence. Arithmetic sequence is a pattern of numbers in which the difference between consecutive terms of the sequence remains constant throughout the sequence. A geometric progression is a sequence of numbers in which any two consecutive terms of the sequence have a common ratio. Harmonic progression is the sequence which forms an arithmetic sequence when the reciprocal of terms are taken in order. 

AM, GM, HM:

AM, GM, HM stands for Arithmetic mean, Geometric mean and Harmonic mean respectively. 

  • AM or Arithmetic Mean is the mean or average of the set of numbers  which is computed by adding all the terms in the set of numbers and dividing the sum by total number of terms. 

  • GM or Geometric Mean is the mean value or the central term in the set of numbers in geometric progression. Geometric mean of a geometric sequence with ‘n’ terms is computed as nth root of the product of all the terms in sequence taken together.

  • HM or Harmonic mean is one of the types of determining the average. Harmonic mean is computed by dividing the number of values in the sequence by the sum of reciprocals of the terms in the sequence.

AM, GM, HM Formula:

Consider a sequence of ‘n’ terms as {a1, a2, a3, a4 …………. An}.

Case 1: If the above sequence is in arithmetic progression, the mean of this sequence is calculated as Arithmetic Mean using the formula.

AM = \[\frac{a1+a2+a3+a4+........+an}{n}\]

Case 2: If the given sequence is geometric progression, the geometric mean of all the terms in the sequence is calculated using the formula.

GM = \[\sqrt[n]{a1\times a2\times a3\times a4\times .....\times an}\]

Case 3: If the sequence is in harmonic progression, the harmonic mean is computed by using the formula.

HM= \[\frac{n}{\frac{1}{a1}+\frac{1}{a2}+\frac{1}{a3}+\frac{1}{a4}+....+\frac{1}{an}}\]

 

Relation Between AM, GM and HM:

Consider two numbers ‘a’ and ‘b’ such that a and b are greater than 0. Terms in the sequence are ‘a’ and ‘b’ and the number of terms in the sequence ‘n = 2’.  If AM GM HM formula is used, AM GM HM can be found as follows.

AM = \[\frac{a+b}{2}\]

GM = \[\sqrt{a+b}\]

HM = \[\frac{2}{\frac{1}{a}+\frac{1}{b}}\] = \[\frac{2}{\frac{b+a}{ab}}\] = \[\frac{2ab}{a+b}\]

Substituting \[\frac{a+b}{2}\] = AM and ab =\[GM^{2}\]

HM = \[\frac{GM^{2}}{AM}\]

The above equation gives the relation between AM,Gm and HM. The equation can also be written as 

AM x HM = \[GM^{2}\]

or

GM = \[\sqrt{AM}\] x HM

AM GM HM in Statistics:

  • AM GM HM in Statistics has a vital role to play in major calculations.

  • Arithmetic Mean being very simple and easy to compute gives one of the measures of the central tendency of a grouped or ungrouped set of data.

  • Geometric mean is used in computation of stock indexes. Also geometric mean is used to calculate the annual returns of the portfolio. Geometric mean is also used in studying biological processes such as cell division and bacterial growth etc.

  • Harmonic mean is used to determine the price earnings ratio and other average multiples in Finance. It is also used in the computation of Fibonacci sequence.

Example Problems:

  1. If five times the geometric mean of two numbers ‘a’ and ‘b’ is equal to the arithmetic mean of those two numbers such that a > b > 0, then compute the value of a + ba - b.

Solution:

        Arithmetic mean of the two numbers is calculated as:

\[AM = \frac{{a + b}}{2}\]

Geometric mean of the two numbers a and b is

        \[GM{\text{ }} = {\text{ }}\sqrt {ab} \]

It is given in the question that Arithmetic mean = 5 times the geometric mean

AM = 5 GM

\[\frac{{a + b}}{2}\] = 5 \[\sqrt{ab}\]

                                    a + b  = 10 \[\sqrt{ab}\]

(a + b)2 = 100 ab

(a - b)2  = (a + b)2 - 4 ab

(a - b)2  = 100 ab - 4 ab

(a - b)2 = 96 ab

a - b = \[\sqrt{96\text{ab}}\]

                                 \[\frac{{a + b}}{{a - b}}\] = \[\sqrt {\frac{{100ab}}{{96ab}}}  = \sqrt {\frac{{25}}{{24}}}  = 1.021\]

 

  1. Find the harmonic mean of two numbers a and b, if their arithmetic mean is 16 and geometric mean is 8 provided that a > b > 0. (Hint: Use relation between AM GM HM formula). Also find the values of a and b.

Solution:

Given: AM = 16 and GM = 8

The relation between AM GM HM is given as:

AM x HM = GM2

16  x  HM = 82 

16 x  HM = 64

\[HM = \frac{{64}}{{16}} = 4\]

 

  To find the numbers:

  Arithmetic mean is given as 

                                     AM =a + b/2

                               16 =a + b/2

a + b  = 32

a = 32 - b

                            Geometric mean is given as

                                    GM = \[\sqrt{ab}\]

                                    8 = \[\sqrt{ab}\]

                                    64 = ab

                                    64 = (32 - b) b

                                    64 = 32b - b2

                                    b2 - 32 b + 64 = 0

 

Fun Facts:

  • It is inferred through a number of calculations and has been proved by experts who use AM GM HM in Statistics that the value of AM is greater than that of GM and HM. The value of GM is greater than that of HM and lesser than that of AM. The value of HM is lesser than that of AM and GM.

AM = \[\frac{a+b}{2}\]

GM = \[\sqrt{ab}\]

HM = \[\frac{2ab}{a+b}\]

  • If zero is one of the terms of a sequence, its geometric mean is zero and the harmonic mean is infinity. 

FAQ (Frequently Asked Questions)

1. What is AM GM HM? Where is it Used?

AM is the short form of arithmetic mean which is the measure of average of the terms in any Mathematical Sequence. Arithmetic mean is used in most of the situations where the mean or average of any statistical data is to be determined.GM is the abbreviation of Geometric Mean which is the measure of nth root of the product of terms in any Mathematical sequence with ‘n’ number of terms. Geometric mean is used in calculations involving growth, investment, determination  of surface areas and volume. Harmonic mean is the measure of ratio of the number of terms in the sequence to the sum of the reciprocals of the term. It is used in the calculations related to speed, production and cost.

2. What is the Inequality Relation between AM GM HM?

The inequality relation between AM GM and HM states that the values of AM GM HM are never equal in most of the cases. Among the three means, arithmetic means generally have the highest value. Geometric mean is greater than harmonic mean. However, its value is lesser than the arithmetic mean. Harmonic mean has the least value among all the three means. The relationship between arithmetic mean, geometric mean and harmonic mean is:

“The product of arithmetic mean and harmonic mean of any two numbers a and b in such a way that a > b > 0 is equal to the square of their geometric mean.”

AM x HM = GM2